A051533 -id:A051533 - cp's OEIS frontend

A185978 Nontriangular numbers which are the sum of two (positive) triangular numbers.

2, 4, 7, 9, 11, 12, 13, 16, 18, 20, 22, 24, 25, 27, 29, 30, 31, 34, 37, 38, 39, 42, 43, 46, 48, 49, 51, 56, 57, 58, 60, 61, 64, 65, 67, 69, 70, 72, 73, 76, 79, 81, 83, 84, 87, 88, 90, 92, 93, 94, 97, 99, 100, 101, 102, 106, 108

Offset: 1

Wolfdieter Lang, Feb 14 2011

This is A051533 (sums of two positive triangular numbers) excluding triangular numbers.
This is also A020756 (sums of two triangular numbers) excluding triangular numbers.
There may be multiple representations, e.g., 16 = 1 + 15 = 6 + 10.

			a(6) = 12 = 6 + 6,
a(17) = 31 = 3 + 28 = 10 + 21.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000217, A020756 (sums of two triangular numbers), A051533 (sums of two positive triangular numbers).

(* First run the program for A051533 *) Complement[%, tri] (* Alonso del Arte, Feb 15 2011 *)

This is the sorted, made unique set {binomial(k+1,2) + binomial(L+1,2), 1 <= k <= L sufficiently large}, excluding members from A000217 (triangular numbers).

A288631 Numbers that are the sum of two nonzero square pyramidal numbers (A000330).

Original entry on oeis.org

2, 6, 10, 15, 19, 28, 31, 35, 44, 56, 60, 69, 85, 92, 96, 105, 110, 121, 141, 145, 146, 154, 170, 182, 195, 205, 209, 218, 231, 234, 259, 280, 286, 290, 295, 299, 315, 340, 344, 376, 386, 390, 399, 408, 415, 425, 440, 476, 489, 507, 511, 520, 525, 536, 561, 570, 589, 597, 646, 651, 655, 664, 670, 680

Offset: 1

Ilya Gutkovskiy, Jun 12 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A000404, A020756, A051533, A102795.

M:= 20: # to get all terms <= A000330(M)
sqp:= [seq(k*(k+1)*(2*k+1)/6, k=1..M)]:
sort(convert(select(`<=`, {seq(seq(sqp[i]+sqp[j], j=1..i),i=1..M-1)},sqp[M]),list)); # Robert Israel, Jun 12 2017

nmax = 700; f[x_] := Sum[x^(k (k + 1) (2 k + 1)/6), {k, 1, 20}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

A180590 Numbers k such that k! is the sum of two triangular numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 15, 16, 17, 21, 24, 27, 28, 29, 32, 33, 34, 42, 49, 54, 59, 66, 68, 72, 79, 80, 81, 85, 86, 95, 96, 99, 102, 109, 118

Offset: 1

Robert G. Wilson v, Sep 10 2010

Numbers k such that there are nonnegative numbers x and y such that x*(x+1)/2 + y*(y+1)/2 = k!. Equivalently, (2x+1)^2 + (2y+1)^2 = 8k! + 2. A necessary and sufficient condition for this is that all the prime factors of 4k!+1 that are congruent to 3 (mod 4) occur to even powers (cf. A001481).
Based on an email from R. K. Guy to the Sequence Fans Mailing List, Sep 10 2010.
See A152089 for further links.

			0! = 1! = T(0) + T(1);
2! = T(1) + T(1);
3! = T(0) + T(3) = T(2) + T(2);
4! = T(2) + T(6);
5! = T(0) + T(15) = T(5) + T(14);
7! = T(45) + T(89);
8! = T(89) + T(269);
9! = T(210) + T(825);
10! = T(665) + T(2610) = T(1770) + T(2030);
13! = T(71504) + T(85680);
15! = T(213384) + T(1603064) = T(299894) + T(1589154);
16! = T(3631929) + T(5353005);
17! = T(12851994) + T(23370945) = T(17925060) + T(19750115);
etc.

Factor Database, Factors of the numbers 4z!+1

Complement of A152089. A171099 gives the number of solutions.

Cf. A000142, A000217, A051533, A076680.

triQ[n_] := IntegerQ@ Sqrt[8 n + 1]; fQ[n_] := Block[{k = 1, lmt = Floor@Sqrt[2*n! ], nf = n!}, While[k < lmt && ! triQ[nf - k (k + 1)/2], k++ ]; r = (Sqrt[8*(nf - k (k + 1)/2) + 1] - 1)/2; Print[{k, r, n}]; If[IntegerQ@r, True, False]]; k = 1; lst = {}; While[k < 69, If[ fQ@ k, AppendTo[lst, k]]; k++ ]; lst

from math import factorial
from itertools import count, islice
from sympy import factorint
def A180590_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(4*factorial(n)+1).items()),count(0))
A180590_list = list(islice(A180590_gen(),15)) # Chai Wah Wu, Jun 27 2022

Edited by N. J. A. Sloane, Sep 24 2010
69 eliminated (see A152089) by N. J. A. Sloane, Sep 24 2010
Extended by Georgi Guninski and D. S. McNeil, Sep 24 2010
a(35)-a(38) from Georgi Guninski, Oct 12 2010
a(39)-a(40) from Tyler Busby, Apr 24 2025

A185979 Numbers which are the sum of two positive triangular numbers in more than one way.

Original entry on oeis.org

16, 31, 42, 46, 51, 56, 72, 76, 81, 94, 106, 111, 121, 123, 126, 133, 141, 146, 156, 157, 172, 174, 181, 186, 191, 196, 198, 211, 216, 225, 226, 231, 237, 241, 246, 256, 259, 268, 276, 281, 286, 289, 291, 297, 301, 306, 310, 315, 321, 326, 328, 331, 336, 342, 346, 354, 361, 366, 367

Offset: 1

Wolfdieter Lang, Feb 15 2011

This is a subsequence of A020756 (sums of two triangular numbers).
This is also a subsequence of A051533 (sums of two positive triangular numbers). This is not a subsequence of A185978 (nontriangular numbers as sums of (positive) triangular numbers). E.g., a(32)=231 is missing there because 231=A000217(21). See A185980.
For the numbers which are sums of two positive triangular numbers in exactly two ways see A064816.
The first number which can be written in exactly three ways as sums of positive triangular numbers is 81.
a(n) gives the positions where A052344 entries are >= 2: A052344(a(n)) >= 2.

			16 = 15 + 1 = 10 + 6.
81 = 45 + 36 = 66 + 15 = 78 + 3.
231= 210 + 21 = 153 + 78

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000217, A020756, A051533, A052344, A064816, A185980.

A287960 Numbers that are the sum of two centered triangular numbers (A005448).

Original entry on oeis.org

2, 5, 8, 11, 14, 20, 23, 29, 32, 35, 38, 41, 47, 50, 56, 62, 65, 68, 74, 77, 83, 86, 89, 92, 95, 104, 110, 113, 116, 119, 128, 131, 137, 140, 146, 149, 155, 167, 170, 173, 176, 182, 185, 194, 197, 200, 203, 209, 212, 218, 221, 230, 236, 239, 245, 251, 254, 263, 266, 272, 275, 278, 281, 284, 293, 299

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

nonn

Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers

Cf. A005448, A020756, A051533, A102795, A286636.

nmax = 300; f[x_] := Sum[x^(3 k (k - 1)/2 + 1), {k, 1, 20}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

8*a(n) = 10+3*A097269(n). - R. J. Mathar, Jul 26 2017

A051693 Square array read by antidiagonals: a(n,k) = sum of two positive regular n-polytopic numbers (i.e., a(n,k) = binomial(x,n) + binomial(y,n) for some x,y >= n).

Original entry on oeis.org

2, 2, 3, 2, 4, 4, 2, 5, 6, 5, 2, 6, 8, 7, 6, 2, 7, 10, 11, 9, 7, 2, 8, 12, 16, 14, 11, 8, 2, 9, 14, 22, 20, 20, 12, 9, 2, 10, 16, 29, 27, 30, 21, 13, 10, 2, 11, 18, 37, 35, 42, 36, 24, 16, 11, 2, 12, 20, 46, 44, 56, 57, 40, 30, 18, 12, 2, 13, 22, 56, 54, 72, 85, 62, 50, 36, 20, 13, 2

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

			a(2,.) = 2,4,6,7,9,11,12,13,16,... = sum of two positive triangular numbers = A051533;
a(3,.) = 2,5,8,11,14,20,21,24,30,36,... = sum of two positive tetrahedral numbers;
First antidiagonals of the array are:
2;
2,3;
2,4,4;
2,5,6,5;
...

Cf. A051533.

nMax = 13; coeff = Floor[nMax/2]+1; row[n_] := Table[Binomial[x, n] + Binomial[y, n], {x, n, coeff*n}, {y, n, coeff*n}] // Flatten // Union; A0 = {}; While[A051693 = Table[row[n][[1 ;; nMax]], {n, 1, nMax}]; A051693 =!= A0, A0 = A051693; coeff++]; Table[A051693[[n-k+1, k]], {n, 1, nMax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 14 2016 *)

A136345 Sums of exactly two positive octagonal numbers A000567.

Original entry on oeis.org

2, 9, 16, 22, 29, 41, 42, 48, 61, 66, 73, 80, 86, 97, 104, 105, 117, 130, 134, 136, 141, 154, 161, 173, 177, 184, 192, 197, 198, 216, 226, 229, 233, 241, 246, 265, 266, 272, 281, 288, 290, 301, 309, 320, 321, 342, 345, 349, 352, 358, 362, 376, 381, 401, 406, 409

Offset: 1

Jonathan Vos Post, Dec 25 2007

For octagonal numbers within this sequence, see: A136346. This is to octagonal numbers A000567 as A051533 is to triangular numbers A000217 and as A000404 is to squares A000290 and as A117104 is to heptagonal numbers A000566.

Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000567, A136117, A136346.

{A000567(i) + A000567(j), i, j > 0}. {i*(3*i-2) + j(3*j-2), i, j > 0}.

Missing term (80) from Giovanni Resta, Jun 19 2016

A185980 Nontriangular numbers that are the sum of two (positive) triangular numbers in more than one way.

Original entry on oeis.org

16, 31, 42, 46, 51, 56, 72, 76, 81, 94, 106, 111, 121, 123, 126, 133, 141, 146, 156, 157, 172, 174, 181, 186, 191, 196, 198, 211, 216, 225, 226, 237, 241, 246, 256, 259, 268, 281, 286, 289, 291, 297, 301, 306, 310, 315, 321, 326, 328, 331, 336, 342, 346, 354

Offset: 1

Wolfdieter Lang, Feb 15 2011

A000217(0)=0, but it will not appear as a summand.

A185979 uses only positive triangular summands, and includes triagonal (or body diagonal) numbers like 231, 276, 406, 666, ...

			16 = 15 + 1 = 10 + 6.
81 = 45 + 36 = 66 + 15 = 78 + 3.

Cf. A000217, A020756 (sums of two triangular numbers), A051533 (sums of two positive triangular numbers), A064816, A185978, A185979.

See the one given for A185978, taking into account multiplicities.

A191765 Integers that are a sum of two nonzero triangular numbers and also the sum of two nonzero square numbers.

Original entry on oeis.org

2, 13, 18, 20, 25, 29, 34, 37, 58, 61, 65, 72, 73, 90, 97, 100, 101, 106, 130, 136, 137, 146, 148, 157, 160, 164, 169, 181, 193, 200, 202, 205, 208, 218, 225, 226, 232, 234, 241, 244, 245, 265, 272, 274, 277, 281, 288, 289, 298, 306, 328, 340, 346, 353, 370, 373, 388, 389, 400

Offset: 1

Ant King, Jun 22 2011

A134422 is a subsequence. - Franklin T. Adams-Watters, Jun 25 2011

			25 is the sum of two nonzero triangular numbers: 10 + 15, and of two nonzero squares: 9 + 16; so 25 is in the sequence.
9 is the sum of two nonzero triangular numbers: 3 + 6, but can be represented as the sum of two squares only using zero: 0 + 9; so 9 is not in the sequence.

P. A. Piza, Problems for Solution: 4425, The American Mathematical Monthly, Vol. 58, No. 2, (February 1951), p. 113.
P. A. Piza, G. W. Walker and C. M. Sandwick, Sr, Problem 4425, The American Mathematical Monthly, Vol. 59, No. 6, (June - July 1952), pp. 417-419.

Cf. A000217, A000290, A191766, intersection of A000404 and A051533, A134422.

data=Length[Reduce[a^2+b^2==1/2 c (c+1)+1/2 d(d+1)== # && a>0 && b>0 && c>0 && d>0,{a,b,c,d},Integers]] &/@Range[400];DeleteCases[Table[If[data[[k]]>0,k,0],{k,1,Length[data]}],0]

A264118 Numbers that can be represented as S1+S2, as T1+T2, and as S3+T3, where S1, S2, S3 are positive squares, and T1, T2, T3 are positive triangular numbers.

Original entry on oeis.org

2, 25, 29, 37, 61, 65, 72, 100, 101, 106, 130, 136, 137, 157, 169, 200, 202, 205, 226, 232, 234, 241, 274, 277, 289, 340, 353, 389, 400, 409, 416, 421, 436, 442, 445, 466, 490, 514, 520, 522, 562, 577, 596, 610, 625, 637, 640, 661, 666, 676, 697, 724, 757, 765, 785, 794, 820

Offset: 1

Alex Ratushnyak, Nov 03 2015

nonn

Intersection of A000404, A051533, A133254.

			a(1) = 2 = 1+1.
25 is a term because 25 = 16 + 9 = 10 + 15 = 4 + 21, and 16, 9, 4 are squares, 10, 15, 21 are triangular numbers.

Cf. A000217, A000290, A000404, A051533, A133254, A264101.

cp's OEIS Frontend

A185978 Nontriangular numbers which are the sum of two (positive) triangular numbers.

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A288631 Numbers that are the sum of two nonzero square pyramidal numbers (A000330).

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A180590 Numbers k such that k! is the sum of two triangular numbers.

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A185979 Numbers which are the sum of two positive triangular numbers in more than one way.

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A287960 Numbers that are the sum of two centered triangular numbers (A005448).

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A051693 Square array read by antidiagonals: a(n,k) = sum of two positive regular n-polytopic numbers (i.e., a(n,k) = binomial(x,n) + binomial(y,n) for some x,y >= n).

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A136345 Sums of exactly two positive octagonal numbers A000567.

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A185980 Nontriangular numbers that are the sum of two (positive) triangular numbers in more than one way.

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A191765 Integers that are a sum of two nonzero triangular numbers and also the sum of two nonzero square numbers.

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