A115173 -id:A115173 - cp's OEIS frontend

A115171 Positive numbers that are not the sum of a triangular number, a square and a positive Fibonacci number.

3970902, 4851691, 7921800, 39022919, 102132857, 110468517, 368495972, 391099413, 395147912, 421129348, 452808398, 608389509, 667231451, 693386307, 698864465, 776218485, 1005771844, 1335351479, 1336652359, 1485470432

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

Comment from Kurt Foster, Jan 20 2009: If one is allowed to use the Fibonacci number zero, then many of these terms disappear. For example, one has 4851691 = 1885^2 + 1611*1612/2 + F_0, 7921800 = 1923^2 + 2906*2907/2 + F_0. See A144642.

Since the number of integers up to B of the form x^2+T_y is about a constant times B/sqrt(log B), heuristics predict that the sequence should be infinite. - Bjorn Poonen, in a posting to the Number Theory List, Dec 22 2008

Donovan Johnson, Table of n, a(n) for n = 1..176 (terms < 4*10^10)

Cf. A115172, A115173, A115174, A115175, A115176, A115177, A144642.

A240088 The number of ways of writing n as an ordered sum of a triangular number (A000217), a square (A000290) and a pentagonal number (A000326).

Original entry on oeis.org

1, 3, 3, 2, 3, 4, 4, 4, 3, 3, 5, 5, 5, 3, 3, 7, 7, 5, 2, 6, 5, 4, 8, 5, 6, 4, 8, 7, 5, 7, 4, 9, 6, 5, 4, 3, 9, 12, 9, 4, 7, 9, 8, 4, 6, 8, 7, 8, 4, 8, 9, 10, 9, 6, 10, 6, 7, 10, 9, 8, 7, 11, 7, 4, 10, 8, 10, 10, 7, 5, 10, 14, 11, 7, 6, 11, 10, 10, 4, 11, 10, 10, 13, 8, 7, 7, 13, 12, 8, 8, 6, 10, 17, 8, 10, 7, 16, 10, 3, 12, 9

Offset: 0

Robert G. Wilson v, Mar 31 2014

nonn

0 and 1 are triangular numbers, square numbers and pentagonal numbers.
It is conjectured that a(n) is always positive - this is one of the conjectures in Conjecture 1.1 of Sun (2009). - N. J. A. Sloane, Apr 01 2014
Note that both the conjecture in A160325 and the conjecture in A160324 imply that a(n) is always positive. - Zhi-Wei Sun, Apr 01 2014
a(n) > 0 for all n < 10^10. - Robert G. Wilson v, Aug 20 2016
Least number to be represented k ways, k >= 1: 0, 3, 1, 5, 10, 19, 15, 22, 31, 51, 61, 37, 82, 71, 126, 96, 92, 136, 162, 187, 206, 276, 191, 261, 236, 247, 317, 302, 401, 292, 422, 547, 456, 544, 551, 612, 591, 577, 521, 666, 742, 726, 682, 877, 796, 1052, 961, 1046, 1171, 1027, ..., . A275999.
Greatest number (conjectured) to be represented k ways, k >= 1: 0, 18, 168, 78, 243, 130, 553, 455, 515, 658, 865, 945, 633, 1918, 2258, 1385, 1583, 2828, 2135, 2335, 2785, 4533, 3168, 3478, 2790, 3868, 4193, 7328, 4953, 5278, 6390, 8148, 8015, 4585, 9160, 10485, 7613, 12333, 12025, 10178, 9923, 9720, 12558, 11340, 17420, 11753, 14893, 16155, 16415, 14343, ..., .
Conjectured lists of numbers that are represented in k >= 1 ways:
1: 0;
2: 3, 18;
3: 1, 2, 4, 8, 9, 13, 14, 35, 98, 168;
4: 5, 6, 7, 21, 25, 30, 34, 39, 43, 48, 63, 78;
5: 10, 11, 12, 17, 20, 23, 28, 33, 69, 193, 203, 230, 243;
6: 19, 24, 32, 44, 53, 55, 74, 90, 111, 130;
7: 15, 16, 27, 29, 40, 46, 56, 60, 62, 68, 73, 84, 85, 95, 108, 113, 123, 135, 139, 163, 165, 273, 553;
8: 22, 26, 42, 45, 47, 49, 59, 65, 83, 88, 89, 93, 112, 119, 125, 134, 140, 144, 186, 205, 233, 244, 320, 405, 455;
9: 31, 36, 38, 41, 50, 52, 58, 100, 109, 124, 160, 214, 249, 308, 358, 515; ..., .

Robert G. Wilson v and Robert Israel, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Science China Mathematics, Vol. 58, No. 7 (2015), 1367-1396; arXiv:0905.0635 [math.NT], 2009-2015 [Edited, _Felix Fröhlich_, Aug 24 2016].
Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes, a message to the Number Theory List, May 8 2009.

Cf. A000925, A008443, A101428, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642.

Cf. A160324, A160325, A160326. - Zhi-Wei Sun, Apr 01 2014

# requires Maple 17 and up
with(SignalProcessing):
N:= 10000;  # to get terms up to a(N)
A:= Array(0..N,datatype=float);
B:= Array(0..N,datatype=float);
C:= Array(0..N,datatype=float);
for i from 0 to floor(sqrt(N)) do A[i^2]:= 1 od:
for i from 0 to floor((1+sqrt(1+8*N))/2) do B[i*(i-1)/2]:= 1 od:
for i from 0 to floor((1+sqrt(1+24*N))/6) do C[i*(3*i-1)/2]:= 1 od:
R:= Convolution(Convolution(A,B),C);
R:= evalhf(map(round,R));
# Note that a(i) = R[i+1] for i from 0 to N
# Robert Israel, Apr 01 2014

p = Table[n (3n - 1)/2, {n, 0, 26}]; s = Table[n^2, {n, 0, 32}]; t = Table[n (n + 1)/2, {n, 0, 45}]; a = Sort@ Flatten@ Table[ p[[i]] + s[[j]] + t[[k]], {i, 26}, {j, 32}, {k, 45}]; Table[ Count[a, n], {n, 0, 105}]

A115174 Numbers that are not the sum of a square, a nonnegative cube and a power of 2.

Original entry on oeis.org

15, 22, 23, 55, 62, 87, 94, 106, 111, 118, 119, 120, 139, 167, 183, 188, 194, 195, 211, 215, 223, 244, 246, 251, 263, 274, 275, 279, 286, 303, 304, 307, 309, 311, 314, 330, 331, 335, 342, 343, 350, 398, 399, 419, 421, 434, 436, 437, 438, 454, 455, 461, 463

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

The square and the cube may be equal to 0, the powers of two starts at 2^0 = 1.

There are 531362099 such numbers up to 10^9, the last one in this range being 999999999.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A115171, A115172, A115173, A115175, A115176, A115177.

N:= 1000: # for terms <= N
S1:= {seq(i^2,i=0..isqrt(N))}:
S2:= {seq(i^3,i=0..floor(N^(1/3)))}:
S3:= {seq(2^i,i=0..ilog2(N))}:
S:= select(`<=`,{seq(seq(seq(a+b+c,a=S1),b=S2),c=S3)},N):
sort(convert({$1..N} minus S,list)); # Robert Israel, Jul 23 2020

Complement[Range[1000], Plus @@@ Tuples[{Range[0, 34]^2, Range[0, 10]^3, 2^Range[0, 9]}]] (* Giovanni Resta, May 02 2016 *)

Name corrected by Giovanni Resta, May 02 2016

A115177 Positive numbers that are not the sum of a triangular number, a nonnegative cube and a positive Fibonacci number.

Original entry on oeis.org

690, 861, 915, 929, 1201, 1319, 1451, 1538, 1578, 1803, 1965, 2104, 2427, 2532, 2573, 2671, 2680, 2734, 2769, 2965, 3001, 3154, 3175, 3255, 3259, 3260, 3336, 3434, 3476, 3603, 3628, 3670, 3805, 3840, 3847, 3898, 3903, 3953, 4050, 4128

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

There are 225943534 such numbers up to 10^9, the last one in this range being 999999997.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A115171, A115172, A115173, A115174, A115175, A115176.

N:=10000: # to get all terms <= N
T:= [seq(t*(t+1)/2, t=0..floor((sqrt(1+8*N)-1)/2))]:
C:= [seq(t^3, t=0..floor(N^(1/3)))]:
F:= [seq(combinat:-fibonacci(t), t=1..floor(log[(sqrt(5)+1)/2](N*sqrt(5))))]:
CF:= select(`<=`,{seq(seq(c+f, c=C),f=F)},N):
sort(convert({$1..N} minus {seq(seq(t+cf,t=T),cf=CF)}, list)); # Robert Israel, Mar 08 2018

Name clarified by Altug Alkan, May 02 2016

A115172 Positive numbers that are not the sum of two triangular numbers and a positive Fibonacci number.

Original entry on oeis.org

926617, 2892173, 9274792, 10991572, 11316512, 19778548, 21269851, 23134703, 23742454, 29833549, 30743348, 30758242, 30829335, 35596922, 35739680, 39313201, 40547231, 52621288, 62966319, 62968936, 66217600, 78581688

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

There are 319 such numbers up to 2*10^9, the last one in this range being 1988099881.

Cf. A115171, A115173, A115174, A115175, A115176, A115177.

A115175 Positive numbers that are not the sum of a triangular number, a fourth power and two positive Fibonacci numbers.

Original entry on oeis.org

1, 787182, 1046727, 1314765, 1586115, 1807851, 1823858, 1834699, 1863616, 1990785, 2018554, 2020769, 2069231, 2120550, 2134796, 2153345, 2176870, 2311342, 2508954, 2540041, 2657052, 2673627, 2730026, 2749536, 2750166

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

The triangular number and the fourth power can be equal to 0.

There are 4445776 such numbers up to 10^9, the last one in this range being 999999850.

Cf. A115171, A115172, A115173, A115174, A115176, A115177.

A115176 Positive numbers that are not the sum of two squares and a positive Fibonacci number.

Original entry on oeis.org

256, 384, 872, 1224, 1512, 1728, 2661, 2675, 2824, 3528, 4656, 5150, 5224, 6016, 6024, 6336, 6592, 6840, 7193, 7528, 7560, 7664, 8016, 8456, 8512, 8952, 8968, 9072, 9696, 9796, 10032, 10300, 10440, 10624, 11000, 11184, 11208, 11256, 11640, 11880

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

There are 2142682 such numbers up to 10^9, the last one in this range being 999999456.

Cf. A115171, A115172, A115173, A115174, A115175, A115177.

isA001481(n)=#bnfisintnorm(bnfinit(z^2+1), n)
is(n)=my(k=1,f); while((f=fibonacci(k++))<=n, if(isA001481(n-f), return(0))); 1 \\ Charles R Greathouse IV, Mar 11 2016

A101428 Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 1, 2, 0, 1, 3, 1, 1, 0, 1, 2, 2, 1, 0, 3, 0, 1, 2, 0, 1, 2, 2, 0, 2, 1, 1, 2, 1, 0, 0, 1, 2, 4, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 0, 2, 1, 1, 2, 1, 1, 2, 1, 1, 0, 2, 0, 2, 0, 0, 4, 1, 1, 2, 0, 0, 4, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 3, 0, 1, 2, 0, 2, 0, 0, 0, 4, 2, 0, 2, 1, 1, 0, 0, 0, 3, 1, 2, 2, 1

Offset: 0

R. J. Mathar, Jan 03 2009

0 is both a triangular number and a square number.
First occurrence of k beginning at 0: 8, 2, 1, 10, 37, 136, 235, 1549, 631, 2314, 2116, 11026, 3997, 148240, 19045, 20827, 25876, 893116, 67951, ?19?, 35974, 187444, 1542655, 354061, 131905, ?25?, ?26?, 835399, 323767, ?29?, 611560, ?31?, 515629, ?33?, ?34?, ?35?, 1187146, ?37?, ?38?, ?39?, 1474939, ..., . - Robert G. Wilson v, Mar 30 2014
Variant of A082660 (which allows only positive triangular numbers). - R. J. Mathar, Apr 28 2020
a(n) is the number of representations of 8*n + 1 as 2*A^2 + B^2 with A even and B odd and both nonnegative integers. - Vladimir Pletser, Aug 30 2025

			Examples: n=1 gives the a(1)=2 cases 1=1+0=0+1; a(26)=2 because 26=25+1=16+10.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000925, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642.

A000217 := proc(n) n*(n+1)/2 ; end:
A101428 := proc(n)
local a,y,t ;
a := 0 ;
for y from 0 do
t := A000217(y) ;
if n-t < 0 then
RETURN(a) ;
else
if issqr(n-t) then
a := a+1 ;
fi;
fi;
od:
end:
for n from 0 to 100 do printf("%a,",A101428(n)) ; od:

t = FoldList[#1 + #2 &, 0, Range@ 15]; s = Range[0, 10]^2, a = Sort@ Flatten@ Table[ s[[j]] + t[[k]], {j, 15}, {k, 11}]; Table[Count[a, n], {n, 0, 104}] (* or *)
triQ[n_] := IntegerQ@ Sqrt[8n + 1]; f[n_] := Block[{c = k = 0, lmt = 2 + Floor[Sqrt[n]]}, While[k < lmt, If[ triQ[n - k^2], c++]; k++]; c]; Array[f, 105, 0] (* Robert G. Wilson v, Mar 30 2014 *)

G.f.: sum(i>=0, x^(i^2) ) * sum(i>=0, x^(i*(i+1)/2) ). - Ralf Stephan, May 17 2014

cp's OEIS Frontend

A115171 Positive numbers that are not the sum of a triangular number, a square and a positive Fibonacci number.

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A240088 The number of ways of writing n as an ordered sum of a triangular number (A000217), a square (A000290) and a pentagonal number (A000326).

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A115174 Numbers that are not the sum of a square, a nonnegative cube and a power of 2.

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A115177 Positive numbers that are not the sum of a triangular number, a nonnegative cube and a positive Fibonacci number.

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A115172 Positive numbers that are not the sum of two triangular numbers and a positive Fibonacci number.

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A115175 Positive numbers that are not the sum of a triangular number, a fourth power and two positive Fibonacci numbers.

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A115176 Positive numbers that are not the sum of two squares and a positive Fibonacci number.

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PARI

A101428 Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290).

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Maple

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