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 A014825 a(n) = 4*a(n-1) + n with n > 1, a(1)=1. 16
 1, 6, 27, 112, 453, 1818, 7279, 29124, 116505, 466030, 1864131, 7456536, 29826157, 119304642, 477218583, 1908874348, 7635497409, 30541989654, 122167958635, 488671834560, 1954687338261, 7818749353066 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. See also Proc. Amer. Math. Soc. 148 (2020), 461-469. Index entries for linear recurrences with constant coefficients, signature (6,-9,4). FORMULA a(n) = (4^(n+1) - 3*n - 4)/9. G.f.: x/((1-4*x)*(1-x)^2). a(n) = Sum_{k=0..n} (n-k)*4^k = Sum_{k=0..n} k*4^(n-k). - Paul Barry, Jul 30 2004 a(n) = Sum{k=0..n} binomial(n+2, k+2)*3^k [Offset 0]. - Paul Barry, Jul 30 2004 a(n) = Sum{k=0..n} Sum_{j=0..2k} (-1)^(j+1)*J(j)*J(2k-j), J(n) = A001045(n). - Paul Barry, Oct 23 2009 Convolution square of A006314. - Michael Somos, Jun 20 2012 E.g.f.: (4*exp(4*x) - (4+3*x)*exp(x))/9. - G. C. Greubel, Feb 18 2020 a(n) = A014916(-n-1)*4^(n+1) = A091919(2*n-2) for all n in Z. - Michael Somos, Oct 02 2020 a(n) = Sum_{k=0..n} A002450(k). - Joseph Brown, May 11 2021 EXAMPLE G.f. = x + 6*x^2 + 27*x^3 + 112*x^4 + 453*x^5 + 1818*x^6 + 7279*x^7 + ... MATHEMATICA RecurrenceTable[{a[1]==1, a[n]==4a[n-1]+n}, a[n], {n, 30}] (* Harvey P. Dale, Oct 12 2011 *) a[ n_]:= SeriesCoefficient[x/((1-4x)(1-x)^2), {x, 0, n}] (* Michael Somos, Jun 20 2012 *) PROG (MAGMA) [(4^(n+1)-3*n-4)/9: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011 (PARI) {a(n) = polcoeff( x / ((1 - x)^2 * (1 - 4*x)) + x * O(x^n), n)} /* Michael Somos, Jun 20 2012 */ (Sage) [(4^(n+1) -3*n -4)/9 for n in (1..30)] # G. C. Greubel, Feb 18 2020 CROSSREFS Cf. A002450 (first differences), A052161 (partial sums). Cf. A001045, A006314, A014916, A053142, A091919. Sequence in context: A005284 A198694 A220101 * A141844 A176476 A079742 Adjacent sequences:  A014822 A014823 A014824 * A014826 A014827 A014828 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified August 2 14:00 EDT 2021. Contains 346424 sequences. (Running on oeis4.)