A113770 -id:A113770 - cp's OEIS frontend

A004770 Numbers of the form 8k+5; or, numbers whose binary expansion ends in 101.

5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93, 101, 109, 117, 125, 133, 141, 149, 157, 165, 173, 181, 189, 197, 205, 213, 221, 229, 237, 245, 253, 261, 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 373, 381, 389, 397, 405, 413, 421, 429, 437, 445

Offset: 1

N. J. A. Sloane

Only numbers of the form 8k+5 may be written as a sum of 5 odd squares. Examples: 5 = 1+1+1+1+1, 13 = 9+1+1+1+1, 21 = 9+9+1+1+1, 29 = 25+1+1+1+1= 9+9+9+1+1, 37 = 9+9+9+9+1 = 25+9+1+1+1, 45 = 25+9+9+1+1=9+9+9+9+9, 53 = 49+1+1+1+1 = 25+25+1+1+1 = 25+9+9+9+1, ... - Philippe Deléham, Sep 03 2005

Positive solutions to the equation x == 5 (mod 8). - K.V.Iyer, Apr 27 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 248.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016813, A017101, A017113, A110534, A113770.

Cf. A004776 (complement), A007521 (primes).

a004770 = (subtract 3) . (* 8)
a004770_list = [5, 13 ..]  -- Reinhard Zumkeller, Aug 17 2012

[8*n-3: n in [1..60]]; // Vincenzo Librandi, May 28 2011

Range[5, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
LinearRecurrence[{2,-1},{5,13},60] (* or *) NestList[#+8&,5,60] (* Harvey P. Dale, Jun 28 2021 *)

a(n)=8*n-3 \\ Charles R Greathouse IV, Sep 24 2015

[8*n-3 for n in range(1,57)] # Stefano Spezia, Jul 23 2025

From R. J. Mathar, Mar 14 2011: (Start)
a(n) = 8*n - 3.
G.f.: x*(5+3*x)/(x-1)^2. (End)
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 28 2011
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(8*x - 3) + 3.
a(n) = A113770(n)/2 = A016813(2*n-1). (End)

A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 4, 2, 0, 0, 1, 5, 6, 6, 2, 0, 0, 1, 6, 8, 10, 8, 2, 0, 0, 1, 7, 10, 14, 16, 12, 2, 0, 0, 1, 8, 12, 18, 24, 26, 16, 2, 0, 0, 1, 9, 14, 22, 32, 42, 42, 24, 2, 0, 0, 1, 10, 16, 26, 40, 58, 72, 68, 32, 2, 0, 0

Offset: 0

Alois P. Heinz, Dec 03 2012

Equivalently, the number of walks of length n-1 on the path graph P_k. - Andrew Howroyd, Apr 17 2017

			A(5,3) = 12: there are 12 words of length 5 over 3-ary alphabet {a,b,c}, where neighboring letters are neighbors in the alphabet: ababa, ababc, abcba, abcbc, babab, babcb, bcbab, bcbcb, cbaba, cbabc, cbcba, cbcbc.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,   1,   1,   1, ...
  0,  1,  2,  3,  4,   5,   6,   7, ...
  0,  0,  2,  4,  6,   8,  10,  12, ...
  0,  0,  2,  6, 10,  14,  18,  22, ...
  0,  0,  2,  8, 16,  24,  32,  40, ...
  0,  0,  2, 12, 26,  42,  58,  74, ...
  0,  0,  2, 16, 42,  72, 104, 136, ...
  0,  0,  2, 24, 68, 126, 188, 252, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0, 2-10 give: A000007, A040000, A029744(n+2) for n>0, A006355(n+3) for n>0, A090993(n+1) for n>0, A090995(n-1) for n>2, A129639, A153340, A153362, A153360.
Rows 0-6 give: A000012, A001477, A005843(k-1) for k>0, A016825(k-2) for k>1, A008590(k-2) for k>2, A113770(k-2) for k>3, A063164(k-2) for k>4.
Main diagonal gives: A102699.
Cf. A198632, A188866, A276562, A208727, A208671.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-1, j, k), j=1..k),
      `if`(i>1, b(n-1, i-1, k), 0)+
      `if`(i b(n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n-1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[iJean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

TransferGf(m,u,t,v,z)=vector(m,i,u(i))*matsolve(matid(m)-z*matrix(m,m,i,j,t(i,j)),vectorv(m,i,v(i)));
ColGf(m,z)=1+z*TransferGf(m, i->1, (i,j)->abs(i-j)==1, j->1, z);
a(n,k)=Vec(ColGf(k,x) + O(x^(n+1)))[n+1];
for(n=0, 7, for(k=0, 7, print1( a(n,k), ", ") ); print(); );
\\ Andrew Howroyd, Apr 17 2017

A188135 a(n) = 8n^2 + 2n + 1.

Original entry on oeis.org

1, 11, 37, 79, 137, 211, 301, 407, 529, 667, 821, 991, 1177, 1379, 1597, 1831, 2081, 2347, 2629, 2927, 3241, 3571, 3917, 4279, 4657, 5051, 5461, 5887, 6329, 6787, 7261, 7751, 8257, 8779, 9317, 9871, 10441, 11027, 11629, 12247, 12881, 13531, 14197, 14879, 15577, 16291, 17021, 17767

Offset: 0

Paul Curtz, Mar 30 2011

Bisection of A193867. - Omar E. Pol, Aug 16 2011

Sequence found by reading the line from 1, in the direction 1, 11, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 04 2011

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A004770, A084849, A113770, A185438, A193867.

[1 + 2*n + 8*n^2: n in [0..50]]; // Vincenzo Librandi, Mar 30 2011

a(n)=8*n^2+2*n+1 \\ Charles R Greathouse IV, Oct 07 2015

First differences: a(n) - a(n-1) = 16*n - 6 = A113770(n) = 2*A004770(n).
Second differences: a(n) - 2*a(n-1) + a(n-2) = 16.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From R. J. Mathar, Apr 06 2011: (Start)
G.f.: -(1+x)*(7*x+1)/(x-1)^3.
a(n) = A084849(2*n). (End)
E.g.f.: exp(x)*(1 + 10*x + 8*x^2). - Elmo R. Oliveira, Oct 19 2024

a(41)-a(47) from Elmo R. Oliveira, Oct 19 2024

A115284 Correlation triangle of 4-C(1,n)-2*C(0,n) (A113311).

Original entry on oeis.org

1, 3, 3, 4, 10, 4, 4, 15, 15, 4, 4, 16, 26, 16, 4, 4, 16, 31, 31, 16, 4, 4, 16, 32, 42, 32, 16, 4, 4, 16, 32, 47, 47, 32, 16, 4, 4, 16, 32, 48, 58, 48, 32, 16, 4, 4, 16, 32, 48, 63, 63, 48, 32, 16, 4, 4, 16, 32, 48, 64, 74, 64, 48, 32, 16, 4

Offset: 0

Paul Barry, Jan 19 2006

Row sums are the coordination sequence for cubic lattice A005899. Diagonal sums are A115285. T(2n,n) is A113770. T(2n,n)-T(2n,n+1) is 1,6,10,10,10,.... (10-4C(1,n)-5C(0,n)).

			Triangle begins:
  1;
  3, 3;
  4, 10, 4;
  4, 15, 15, 4;
  4, 16, 26, 16, 4;
  4, 16, 31, 31, 16, 4;
  4, 16, 32, 42, 32, 16, 4;

G.f.: (1+x)^2*(1+x*y)^2/((1-x)*(1-x*y)*(1-x^2*y)).

T(n, k) = Sum_{j=0..n} [j<=k]*(4-C(1, k-j)-2*C(0, k-j))*[j<=n-k]*(4-C(1, n-k-j)-2*C(0, n-k-j)).

A247961 Numbers n such that 45^n + 2 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 38, 40, 104, 114, 135, 417, 1251, 14786, 16720, 43831, 152659

Offset: 1

Vincenzo Librandi, Oct 02 2014

Terms of A016969 and A113770 (except 1) are not in the sequence. - Bruno Berselli, Oct 02 2014

a(17) > 2*10^5. - Robert Price, Sep 15 2015

Henri & Renaud Lifchitz, PRP Top Records, Search for 45^n + 2.

Cf. similar sequences listed in A247957.

[n: n in [0..400]| IsPrime( 45^n + 2 )];

A247961:=n->`if`(isprime(45^n+2),n,NULL): seq(A247961(n),n=1..500); # Wesley Ivan Hurt, Oct 02 2014

Select[Range[0, 2000], PrimeQ[45^# + 2] &]

is(n)=ispseudoprime(45^n+2) \\ Charles R Greathouse IV, Jun 06 2017

a(13)-a(16) from Robert Price, Sep 15 2015

A268730 a(n) = Product_{k = 0..n} 2(8k + 5).

Original entry on oeis.org

10, 260, 10920, 633360, 46868640, 4218177600, 447126825600, 54549472723200, 7527827235801600, 1159285394313446400, 197078517033285888000, 36656604168191175168000, 7404634041974617383936000, 1614210221150466589698048000, 377725191749209181989343232000

Offset: 0

Ilya Gutkovskiy, Feb 12 2016

			a(0) = (1 + 2 + 3 + 4) = 10;
a(1) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) = 260;
a(2) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) *(9 + 10 + 11 + 12) = 10920;
a(3) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) *(9 + 10 + 11 + 12)*(13 + 14 + 15 + 16) = 633360, etc.

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A000027, A113770, A147630.

[&*[(16*k+10): k in [0..n-1]]: n in [1..20]]; // Vincenzo Librandi, Feb 12 2016

FullSimplify[Table[(2^(4 n + 13/4) Gamma[1/8] Gamma[n + 13/8])/(Sqrt[Pi] Gamma[1/4]), {n, 0, 14}]]
Table[Product[16 k + 10, {k, 0, n - 1}], {n, 20}] (* Vincenzo Librandi, Feb 12 2016 *)

x='x+O('x^50); Vec(serlaplace(10/(1 - 16*x)^(13/8))) \\ G. C. Greubel, Apr 09 2017

a(n) = (2^(4*n + 13/4)*Gamma(1/8)*Gamma(n + 13/8))/(sqrt(Pi)*Gamma(1/4)), where Gamma(x) is the gamma function.
a(n) = 2*(8*n + 5)*a(n - 1), a(0)=10.
Sum_{n>=0} 1/a(n) = (exp(1/16)*(Gamma(5/8) - Gamma(5/8, 1/16)))/(2*sqrt(2)) = 0.10393932939417..., where Gamma(a, x) is the incomplete gamma function.
a(n) ~ sqrt(Pi) * 2^(4*n+9/2) * n^(n+9/8) / (Gamma(5/8) * exp(n)). - Vaclav Kotesovec, Feb 20 2016
G.f.: 10/(1-b(1)x/(1-(b(1)-10)x/(1-b(2)x/(1-(b(2)-10)x/(1-b(3)x/(...)))))), where b(n)=2(5+8n), i.e. 26,42,58,74. - Benedict W. J. Irwin, Feb 24 2016
a(n) = 2^(n+1)*A147625(n+2). - R. J. Mathar, Jun 07 2016
E.g.f.: 10/(1 - 16*x)^(13/8). - Ilya Gutkovskiy, Jun 07 2016

cp's OEIS Frontend

A004770 Numbers of the form 8k+5; or, numbers whose binary expansion ends in 101.

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A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A188135 a(n) = 8*n^2 + 2*n + 1.

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A115284 Correlation triangle of 4-C(1,n)-2*C(0,n) (A113311).

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A247961 Numbers n such that 45^n + 2 is prime.

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Magma

Maple

Mathematica

PARI

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A268730 a(n) = Product_{k = 0..n} 2*(8*k + 5).

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Magma

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Formula

A188135 a(n) = 8n^2 + 2n + 1.

A268730 a(n) = Product_{k = 0..n} 2(8k + 5).